PRIMES: ML-Enhanced Coupling and PDDO-Enhanced ML Approaches for Complex Problems

This project aims to enhance the research and training capacity in scientific machine learning (SciML) for undergraduate students at Texas A&M University-San Antonio (A&M-SA), a Hispanic serving and primarily undergraduate institution. Through engagement with the Institute for Mathematical and Statistical Innovation (IMSI), the project will cultivate research collaborations in machine learning (ML).

Primary goals are: (1) formalize a partnership between A&M-SA and IMSI, including the PI's participation in the Spring 2025 long program on Uncertainty Quantification and Artificial Intelligence for Complex Systems, (2) advance the PI's scholarship and undergraduate training through research opportunities, curriculum development, and dissemination of content material in SciML. The project will transform the instructional and training ecosystem at A&M-SA especially supporting SciML.

The PI would be building strong computational and coding skills for undergraduate students to be competitive in the marketplace and ready to join the STEM workforce.

The project's first research thrust is ML-enhanced iterative coupling of complex local-to-nonlocal (LtN) problems. While local problems enjoy computational feasibility, they lose their effectiveness when dealing with challenging physics such as fracture. In numerous applications, fracture is localized in a region.

This is where a nonlocal method such as peridynamics (PD) should take over to capture the physics, nonetheless with heavy computational cost. This is the rationale of LtN coupling. Hence, coupling approaches will make such problems computationally realistic and feasible.

The project’s second research thrust is the peridynamic differential operator (PDDO)-enhanced ML approaches to solve complex fluid flow problems. PDDO enables numerical differentiation through integration by converting differentiation to its nonlocal PD (integral) form. It maintains the ability to use partial differential equations for modeling while treating discontinuities seamlessly in an integral representation.

Its strength lies in capturing sharp gradients, even gradient singularities. The performance of existing physics guided ML approaches may degrade in the presence of sharp gradients. This can be remedied by incorporating nonlocal interactions into the neural network input together with local space and time variables.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.